View source for Icosidodecahedron

{{short description|Archimedean solid with 32 faces}}
{{infobox polyhedron
 | name = Icosidodecahedron
 | image = Icosidodecahedron (green).png
 | type = [[Archimedean solid]]<br>[[Uniform polyhedron]]<br>[[Quasiregular polyhedron]]
 | faces = 32
 | edges = 60
 | vertices = 30
 | symmetry = [[Icosahedral symmetry]] {{math|I<sub>h</sub>}}
 | angle = 142.62°
 | dual = [[Rhombic triacontahedron]]
 | properties = [[Convex set|convex]]
 | vertex_figure = Polyhedron 12-20 vertfig.svg
 | net = Polyhedron 12-20 net.svg
}}
[[File:Icosidodecahedron.stl|thumb|3D model of an icosidodecahedron]]

In [[geometry]], an '''icosidodecahedron''' or '''pentagonal gyrobirotunda''' is a [[polyhedron]] with twenty (''icosi-'') [[triangular]] faces and twelve (''dodeca-'') [[pentagon]]al faces. An icosidodecahedron has 30 identical [[Vertex (geometry)|vertices]], with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the [[Archimedean solid]]s and more particularly, a [[quasiregular polyhedron]].

== Construction ==
One way to construct the icosidodecahedron is to start with two [[pentagonal rotunda]] by attaching them to their bases. These rotundas cover their decagonal base so that the resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to one of the [[Johnson solids]], the [[pentagonal orthobirotunda]]. The difference is that the icosidodecahedron is constructed by twisting its rotundas by 36°, a process known as [[gyration]], resulting in the pentagonal face connecting to the triangular one. The icosidodecahedron has an alternative name, ''pentagonal gyrobirotunda''.{{r|berman|os}}
{{multiple image
 | image1 = Icosidodecahedron.png
 | image2 = Dissected icosidodecahedron.png
 | image3 = Pentagonal orthobirotunda solid.png
 | footer = The difference between icosidodecahedron and pentagonal orthobirotunda, and its dissection.
 | align = center
 | total_width = 400
}}

There is another way to construct it, and that is [[Rectification (geometry)|rectification]] of an [[icosahedron]] or a [[dodecahedron]].

Convenient [[Cartesian coordinates]] for the vertices of an icosidodecahedron with unit edges are given by the [[even permutation]]s of:
<math display="block"> (\pm 1, 0, 0), \qquad \tfrac{1}{2}\left(\pm \varphi, \pm \tfrac{1}{\varphi}, \pm 1 \right), </math>
where <math> \varphi </math> denotes the [[golden ratio]].{{r|dm}}

== Properties ==
The surface area of an icosidodecahedron {{mvar|A}} can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron {{mvar|V}} can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:{{r|berman}}
<math display="block">\begin{align}
A &= \left(5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right) a^2 &\approx 29.306a^2 \\
V &= \frac{45+17\sqrt{5}}{6}a^3 &\approx 13.836a^3.
\end{align}</math>

The [[dihedral angle]] of an icosidodecahedron between pentagon-to-triangle is
<math display="block"> \arccos \left(-\sqrt{\frac{5 + 2\sqrt{5}}{15}} \right) \approx 142.62^\circ, </math>
determined by calculating the angle of a pentagonal rotunda.{{r|williams}}

An icosidodecahedron has [[icosahedral symmetry]], and its first [[stellation]] is the [[Compound of dodecahedron and icosahedron|compound]] of a [[dodecahedron]] and its dual [[icosahedron]], with the vertices of the icosidodecahedron located at the midpoints of the edges of either.

The icosidodecahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.{{r|diudea}} The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the [[vertex figure]] of an icosidodecahedron is <math> (3 \cdot 5)^2 = 3^2 \cdot 5^2 </math>. Its [[dual polyhedron]] is [[rhombic triacontahedron]], a [[Catalan solid]].{{r|williams}}

{{multiple image
 | align = right  | total_width = 400
 | image1 = Polyhedron 12-20, davinci.png
 | image2 = Spherical icosidodecahedron with colored cicles.png
 | footer = The 60 edges form 6 [[decagon]]s corresponding to [[great circle]]s in the spherical tiling.
}}
The icosidodecahedron has 6 central [[decagon]]s. Projected into a sphere, they define 6 [[great circle]]s. {{harvtxt|Fuller|1975}} used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his [[31 great circles of the spherical icosahedron]].{{r|fuller}}

The long radius (center to vertex) of the icosidodecahedron is in the [[golden ratio]] to its edge length; thus its radius is {{mvar|φ}} if its edge length is 1, and its edge length is {{math|{{sfrac|1|''φ''}}}} if its radius is 1.{{r|williams}} Only a few uniform polytopes have this property, including the four-dimensional [[600-cell]], the three-dimensional icosidodecahedron, and the two-dimensional [[Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.) These ''radially golden'' polytopes can be constructed, with their radii, from [[Golden triangle (mathematics)|golden triangle]]s which meet at the center, each contributing two radii and an edge.

==Related polytopes==
The icosidodecahedron is a [[Rectification (geometry)|rectified]] [[dodecahedron]] and also a rectified [[icosahedron]], existing as the full-edge truncation between these regular solids.

The icosidodecahedron contains 12 pentagons of the [[dodecahedron]] and 20 triangles of the [[icosahedron]]:
{{Icosahedral truncations}}

The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with [[vertex configuration]]s (3.''n'')<sup>2</sup>, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With [[orbifold notation]] symmetry of *''n''32 all of these tilings are [[wythoff construction]] within a [[fundamental domain]] of symmetry, with generator points at the right angle corner of the domain.<ref>[[Harold Scott MacDonald Coxeter|Coxeter]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', Third edition, (1973), Dover edition, {{ISBN|0-486-61480-8}} (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)</ref><ref>[http://www.google.com/search?q=Two-Dimensional+Symmetry+Mutation ''Two Dimensional symmetry Mutations'' by Daniel Huson]</ref>
{{Quasiregular3 small table}}

{{Quasiregular5 table}}

=== Related polyhedra ===
[[File:Icosidecahedron in truncated cube.png|120px|thumb|A topological icosi&shy;dodeca&shy;hedron in truncated cube, inserting 6 vertices in center of octagons, and dissecting them into 2 pentagons and 2 triangles.]]
The [[truncated cube]] can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has [[pyritohedral symmetry]].

Eight [[uniform star polyhedron|uniform star polyhedra]] share the same [[vertex arrangement]]. Of these, two also share the same [[edge arrangement]]: the [[small icosihemidodecahedron]] (having the triangular faces in common), and the [[small dodecahemidodecahedron]] (having the pentagonal faces in common). The vertex arrangement is also shared with the [[uniform polyhedron compound|compounds]] of [[compound of five octahedra|five octahedra]] and of [[compound of five tetrahemihexahedra|five tetrahemihexahedra]].
{|class="wikitable" width="400" style="vertical-align:top;text-align:center"
|align=center|[[Image:Icosidodecahedron.png|100px]]<br>Icosidodecahedron
|align=center|[[Image:Small icosihemidodecahedron.png|100px]]<br>[[Small icosihemidodecahedron]]
|align=center|[[Image:Small dodecahemidodecahedron.png|100px]]<br>[[Small dodecahemidodecahedron]]
|-
|align=center|[[Image:Great icosidodecahedron.png|100px]]<br>[[Great icosidodecahedron]]
|align=center|[[Image:Great dodecahemidodecahedron.png|100px]]<br>[[Great dodecahemidodecahedron]]
|align=center|[[Image:Great icosihemidodecahedron.png|100px]]<br>[[Great icosihemidodecahedron]]
|-
|align=center|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|align=center|[[Image:Small dodecahemicosahedron.png|100px]]<br>[[Small dodecahemicosahedron]]
|align=center|[[Image:Great dodecahemicosahedron.png|100px]]<br>[[Great dodecahemicosahedron]]
|-
|align=center|[[Image:Compound of five octahedra.png|100px]]<br>[[Compound of five octahedra]]
|align=center|[[Image:UC18-5 tetrahemihexahedron.png|100px]]<br>[[Compound of five tetrahemihexahedra]]
|}

=== Related polychora ===

In four-dimensional geometry, the icosidodecahedron appears in the [[Regular polytopes|regular]] [[600-cell]] as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed [[hypersphere]] from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wireframe figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron.

If a [[600-cell]] is [[Stereographic projection|stereographically projected]] to 3-space about any vertex and all points are normalised, the [[geodesic]]s upon which edges fall comprise the icosidodecahedron's [[Barycentric subdivision#Barycentric subdivision of a convex polytope|barycentric subdivision]].

== Graph ==
{| class=wikitable align=right width=360
!Icosidodecahedral graph
!Hemi-icosidodecahedral graph
|- valign=top
|[[File:Icosidodecahedral graph.png|180px]]
|[[File:Hemi-icosidodecahedral-graph.png|180px]]<BR>Perimeter opposite edges and vertices glued
|}
The [[Skeleton (topology)|skeleton]] of an icosidodecahedron can be represented as the symmetric [[Graph (discrete mathematics)|graph]] with 30 [[Vertex (graph theory)|vertices]] and 60 edges, one of the [[Archimedean graph]]s. It is a symmetric [[quartic graph]], meaning that each vertex is connected to four other vertices.<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|author-link2=Robin Wilson (mathematician)|title=An Atlas of Graphs|publisher=[[Oxford University Press]]|year= 1998|page=269}}</ref>

The related ''hemi-icosidodecahederal graph'' exists in the [[real projective plane]], with 15 vertices and 30 edges. It is also a symmetric quartic graph. It can be drawn inside of a regular decagon perimeter, with opposite vertices and edges glued together.<ref>https://www.weddslist.com/rmdb/1graph.php?gr=hidh</ref>

== Applications ==
[[File:hoberman_sphere.gif|thumb|Animation of a [[Hoberman sphere]] in the form of a spherical icosidodecahedron with each of its six [[geodesic]]s distinctly coloured, the grey ring viewed face on and the far hemisphere muted]]
The icosidodecahedron may appear in structures, as in the geodesic dome or the [[Hoberman sphere]].

Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31 [[COPII]] coat-protein formations.{{r|rs}}

The icosidodecahedron may also found in popular culture. In [[Star Trek|Star Trek universe]], the Vulcan game of logic [[List_of_games_in_Star_Trek#Kal-toh|Kal-Toh]] has the goal of creating a shape with two nested [[holographic]] icosidodecahedra joined at the midpoints of their segments.

==See also==
*[[Cuboctahedron]]
*[[Great truncated icosidodecahedron]]
*[[Icosahedron]]
*[[Rhombicosidodecahedron]]
*[[Truncated icosidodecahedron]]

== References ==
<references>
<ref name=berman>{{cite journal
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}</ref>

<ref name=diudea>{{cite book
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39
 | page = 39
}}</ref>

<ref name=dm>{{cite book
 | last1 = de Graef | first1 = Marc
 | last2 = McHenry | first2 = Michael
 | title = Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry
 | publisher = Cambridge University Press
 | url = https://books.google.com/books?id=NMUgAwAAQBAJ&pg=PA500
 | page = 500
 | year = 2012
 | edition = 2nd
| isbn = 978-1-139-56047-4
 }}</ref>

<ref name=fuller>{{cite book
 | last = Fuller | first = R. B. | authorlink = R. Buckminster Fuller
 | title = Synergetics: Explorations in the Geometry of Thinking
 | year = 1975
 | publisher = MacMillan
 | url = https://books.google.com/books?id=AKDgDQAAQBAJ&dq=31+great+circles&pg=PA183
 | page = 183&ndash;185
 | isbn = 978-0-02-065320-2
}}</ref>

<ref name=os>{{cite book
 | last1 = Ogievetsky | first1 = O.
 | last2 = Shlosman | first2 = S.
 | editor-last1 = Novikov | editor-first1 = S.
 | editor-last2 = Krichever | editor-first2 = I.
 | editor-last3 = Ogievetsky | editor-first3 = O.
 | editor-last4 = Shlosman | editor-first4 = S.
 | year = 2021
 | title = Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry
 | contribution = Platonic compounds and cylinders
 | url = https://books.google.com/books?id=UsspEAAAQBAJ&pg=PA477
 | page = 477
 | publisher = [[American Mathematical Society]]
 | isbn = 978-1-4704-5592-7
}}</ref>

<ref name=rs>{{cite journal 
 | last1 = Russell | first1 = Christopher
 | last2 = Stagg | first2 = Scott
 | title = New Insights into the Structural Mechanisms of the COPII Coat
 | journal = Traffic
 | date = 11 February 2010
 | volume = 11 | issue = 3 | pages = 303–310
 | doi = 10.1111/j.1600-0854.2009.01026.x
 | pmid = 20070605
 | doi-access = free
}}</ref>

<ref name=williams>{{cite book
 | last = Williams | first = Robert | authorlink = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | isbn = 978-0-486-23729-9
 | url = https://archive.org/details/geometricalfound00will/page/86/mode/1up?view=theater
 | page = 86
}}</ref>
</references>

*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}

==External links==
*{{mathworld2 |urlname=Icosidodecahedron |title=Icosidodecahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x5o - id}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1D1NiLVqdeGtmgRYVoqDtRVdULYX8yuwCc3QmkefHkBBQFfG061UUcmoeyIbYKETw4p9pLg3WecHMPxDRy8JGaGKHlYWQJCqLbA3yu66NU9ZKOdvejvHBGtMuZp4O89LBszvnEgWNFbU7I2JwhnVp8hQBn8wSw1ifNCAzgGapWzCkvOVoJPQgcrWxuNXlSA1ESwiCSBVdqj7pKquEaYsKQyb&name=Icosidodecahedron#applet Editable printable net of an icosidodecahedron with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

{{Polyhedra}}
{{Archimedean solids}}
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[[Category:Archimedean solids]]
[[Category:Quasiregular polyhedra]]